$\kappa$-General-Relativity I: a Non-Commutative GR Theory with the $\kappa$-Minkowski Spacetime as its Flat Limit

TL;DR

This paper develops a non-commutative general relativity theory using twist deformation of diffeomorphisms on kappa-Minkowski spacetime, aiming to incorporate quantum gravitational effects into a covariant framework.

Contribution

It introduces a novel covariant gravitational theory based on deforming the IGL(3,1) group, avoiding issues with Poincare twisting, and establishes a consistent classical limit via Inonu-Wigner contraction.

Findings

Constructed a covariant non-commutative gravity model

Demonstrated the relativistic consistency of twisted kappa-Minkowski spacetime

Proposed a classical limit procedure for the theory

Abstract

We employ a twist deformation of infinitesimal diffeomorphisms to construct a modification of General Relativity on a non-commutative spacetime extending the local kappa-Minkowski geometry. This spacetime arises in Deformed Special Relativity (DSR) models, where a fundamental length scale is incorporated into Special Relativity as an effective description of quantum gravitational effects. To avoid the mathematical and physical inconsistencies associated with twisting the Poincare group, we instead deform the dilatation-enlarged IGL(3,1) group, constructing a covariant and explicitly consistent gravitational theory (distinct from Weyl gravity). The relativistic consistency of the twisted kappa-Minkowski spacetime is demonstrated, including deformed transformations and differential structures. A physically motivated Inonu-Wigner (IW) contraction procedure is suggested to enable a well-defined classical limit, addressing the correspondence issue. This framework provides a consistent foundation for a dynamical sector of DSR and allows, in future treatment, explicit computations that could advance phenomenological predictions.